Neeb KH (2014)
Publication Type: Book chapter / Article in edited volumes
Publication year: 2014
Publisher: Springer
Edited Volumes: Lie theory workshops
Series: Developmenst in Mathematics
Book Volume: 37
Pages Range: 197 - 243
ISBN: 978-3-319-09933-0
URI: https://arxiv.org/abs/1308.1500
DOI: 10.1007/978-3-319-09934-7_8
In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group U(H)" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">U(H) of a real, complex or quaternionic separable Hilbert space and the subgroup U∞(H)" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">U∞(H), consisting of those unitary operators g for which g −1 is compact. The Kirillov–Olshanski theorem on the continuous unitary representations of the identity component U∞(H)0" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">U∞(H)0 asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalized to inseparable spaces. These results are carried over to the full unitary group by Pickrell’s theorem, asserting that the separable unitary representations of U(H)" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">U(H), for a separable Hilbert space H" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">H, are uniquely determined by their restriction to U∞(H)0" id="MathJax-Element-6-Frame" role="presentation" style="position: relative;" tabindex="0">U∞(H)0. For the 10 classical infinite rank symmetric pairs (G, K) of non-unitary type, such as (GL(H),U(H))" id="MathJax-Element-7-Frame" role="presentation" style="position: relative;" tabindex="0">(GL(H),U(H)), we also show that all separable unitary representations are trivial.
APA:
Neeb, K.H. (2014). Unitary representations of Unitary Groups. In G. Mason, I. Penkov, J. Wolf (Eds.), Lie theory workshops. (pp. 197 - 243). Springer.
MLA:
Neeb, Karl Hermann. "Unitary representations of Unitary Groups." Lie theory workshops. Ed. G. Mason, I. Penkov, J. Wolf, Springer, 2014. 197 - 243.
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